This document provides an overview of reinforcement learning concepts. It introduces reinforcement learning as using rewards to learn how to maximize utility. It describes Markov decision processes (MDPs) as the framework for modeling reinforcement learning problems, including states, actions, transitions, and rewards. It discusses solving MDPs by finding optimal policies using value iteration or policy iteration algorithms based on the Bellman equations. The goal is to learn optimal state values or action values through interaction rather than relying on a known model of the environment.

1. Reinforcement
Learning
Slides based on those used in Berkeley's AI class taught by Dan Klein

2. Reinforcement Learning
 Basic idea:
 Receive feedback in the form of rewards
 Agent’s utility is defined by the reward function
 Must (learn to) act so as to maximize expected rewards

3. Grid World
 The agent lives in a grid
 Walls block the agent’s path
 The agent’s actions do not always
go as planned:
 80% of the time, the action North
takes the agent North
(if there is no wall there)
 10% of the time, North takes the
agent West; 10% East
 If there is a wall in the direction the
agent would have been taken, the
agent stays put
 Small “living” reward each step
 Big rewards come at the end
 Goal: maximize sum of rewards*

4. Grid Futures
Deterministic Grid World Stochastic Grid World
X
X
E N S W
X
E N S W
?
X
4
X X

5. Markov Decision Processes
 An MDP is defined by:
 A set of states s  S
 A set of actions a  A
 A transition function T(s,a,s’)
 Prob that a from s leads to s’
 i.e., P(s’ | s,a)
 Also called the model
 A reward function R(s, a, s’)
 Sometimes just R(s) or R(s’)
 A start state (or distribution)
 Maybe a terminal state
 MDPs are a family of non-
deterministic search problems
 Reinforcement learning: MDPs
where we don’t know the
transition or reward functions
5

6. Keepaway
6
 http://www.cs.utexas.edu/~AustinVilla/sim/
keepaway/swf/learn360.swf
 SATR
 S0, S0

7. What is Markov about MDPs?
 Andrey Markov (1856-1922)
 “Markov” generally means that given
the present state, the future and the
past are independent
 For Markov decision processes,
“Markov” means:

8. Solving MDPs
 In deterministic single-agent search problems, want an
optimal plan, or sequence of actions, from start to a goal
 In an MDP, we want an optimal policy *: S → A
 A policy  gives an action for each state
 An optimal policy maximizes expected utility if followed
 Defines a reflex agent
Optimal policy when
R(s, a, s’) = -0.03 for all
non-terminals s

9. Example Optimal Policies
R(s) = -2.0
R(s) = -0.4
R(s) = -0.03
R(s) = -0.01
9

10. MDP Search Trees
 Each MDP state gives an expectimax-like search tree
a
10
s
s’
s, a
(s,a,s’) called a transition
T(s,a,s’) = P(s’|s,a)
R(s,a,s’)
s,a,s’
s is a state
(s, a) is a
q-state

11. Utilities of Sequences
 In order to formalize optimality of a policy, need to
understand utilities of sequences of rewards
 Typically consider stationary preferences:
 Theorem: only two ways to define stationary utilities
 Additive utility:
 Discounted utility:
11

12. Infinite Utilities?!
 Problem: infinite state sequences have infinite rewards
 Solutions:
 Finite horizon:
 Terminate episodes after a fixed T steps (e.g. life)
 Gives nonstationary policies ( depends on time left)
 Absorbing state: guarantee that for every policy, a terminal state
will eventually be reached
 Discounting: for 0 < < 1
 Smaller means smaller “horizon” – shorter term focus
12

13. Discounting
 Typically discount
rewards by < 1
each time step
 Sooner rewards
have higher utility
than later rewards
 Also helps the
algorithms
converge
13

14. Recap: Defining MDPs
 Markov decision processes:
 States S
0
 Start state s
 Actions A
 Transitions P(s’|s,a) (or T(s,a,s’))
 Rewards R(s,a,s’) (and discount )
 MDP quantities so far:
 Policy = Choice of action for each state
 Utility (or return) = sum of discounted rewards
s
a
s, a
s,a,s’
s’
14

15. Optimal Utilities
 Fundamental operation: compute
the values (optimal expectimax
utilities) of states s
 Why? Optimal values define
optimal policies!
 Define the value of a state s:
V*(s) = expected utility starting in s
and acting optimally
 Define the value of a q-state (s,a):
Q*(s,a) = expected utility starting in s,
taking action a and thereafter
acting optimally
 Define the optimal policy:
*(s) = optimal action from state s
s
a
s, a
s,a,s’
s’
15

16. The Bellman Equations
 Definition of “optimal utility” leads to a
simple one-step lookahead relationship
amongst optimal utility values:
Optimal rewards = maximize over first
action and then follow optimal policy
 Formally:
s
a
s, a
s,a,s’
s’
16

17. Solving MDPs
 We want to find the optimal policy *
 Proposal 1: modified expectimax search, starting from
each state s:
s
a
s, a
s,a,s’
s’
17

18. Why Not Search Trees?
 Why not solve with expectimax?
 Problems:
 This tree is usually infinite (why?)
 Same states appear over and over (why?)
 We would search once per state (why?)
 Idea: Value iteration
 Compute optimal values for all states all at
once using successive approximations
 Will be a bottom-up dynamic program
similar in cost to memoization
 Do all planning offline, no replanning
needed!
18

19. Value Estimates
 Calculate estimates V *(s)
k
 Not the optimal value of s!
 The optimal value
considering only next k
time steps (k rewards)
 As k  , it approaches
the optimal value
 Almost solution: recursion
(i.e. expectimax)
 Correct solution: dynamic
programming
19

20. Value Iteration
 Idea:
 Start with V *(s) = 0, which we know is right (why?)
0
 Given Vi
*, calculate the values for all states for depth i+1:
 This is called a value update or Bellman update
 Repeat until convergence
 Theorem: will converge to unique optimal values
 Basic idea: approximations get refined towards optimal values
 Policy may converge long before values do
20

21. Example: 
=0.9, living
reward=0, noise=0.2
Example: Bellman Updates
max happens for
a=right, other
actions not shown
21

22. Example: Value Iteration
 Information propagates outward from terminal
states and eventually all states have correct
value estimates
V2 V3
22

23. Convergence*
 Define the max-norm:
 Theorem: For any two approximations U and V
 I.e. any distinct approximations must get closer to each other,
so, in particular, any approximation must get closer to the true U
and value iteration converges to a unique, stable, optimal
solution
 Theorem:
 I.e. once the change in our approximation is small, it must also
be close to correct
23

24. Practice: Computing Actions
 Which action should we chose from state s:
 Given optimal values V?
 Given optimal q-values Q?
 Lesson: actions are easier to select from Q’s!
24

25. Utilities for Fixed Policies
 Another basic operation: compute
the utility of a state s under a fix
(general non-optimal) policy
 Define the utility of a state s, under a
fixed policy :
V(s) = expected total discounted
rewards (return) starting in s and
following 
 Recursive relation (one-step look-
ahead / Bellman equation):
s
(s)
s, (s)
s, (s),s’
s’
26

26. Value Iteration
 Idea:
 Start with V *(s) = 0, which we know is right (why?)
0
 Given Vi
*, calculate the values for all states for depth i+1:
 This is called a value update or Bellman update
 Repeat until convergence
 Theorem: will converge to unique optimal values
 Basic idea: approximations get refined towards optimal values
 Policy may converge long before values do
27

27. Policy Iteration
29
 Problem with value iteration:
 Considering all actions each iteration is slow: takes |A| times longer
than policy evaluation
 But policy doesn’t change each iteration, time wasted
 Alternative to value iteration:
 Step 1: Policy evaluation: calculate utilities for a fixed policy (not optimal
utilities!) until convergence (fast)
 Step 2: Policy improvement: update policy using one-step lookahead
with resulting converged (but not optimal!) utilities (slow but infrequent)
 Repeat steps until policy converges
 This is policy iteration
 It’s still optimal!
 Can converge faster under some conditions

28. Policy Iteration
 Policy evaluation: with fixed current policy , find values
with simplified Bellman updates:
 Iterate until values converge
 Policy improvement: with fixed utilities, find the best
action according to one-step look-ahead
30

29. Comparison
31
 In value iteration:
 Every pass (or “backup”) updates both utilities (explicitly, based
on current utilities) and policy (possibly implicitly, based on
current policy)
 In policy iteration:
 Several passes to update utilities with frozen policy
 Occasional passes to update policies
 Hybrid approaches (asynchronous policy iteration):
 Any sequences of partial updates to either policy entries or
utilities will converge if every state is visited infinitely often

30. Reinforcement Learning
36
 Reinforcement learning:
 Still assume an MDP:
 A set of states s  S
 A set of actions (per state) A
 A model T(s,a,s’)
 A reward function R(s,a,s’)
 Still looking for a policy (s)
 New twist: don’t know T or R
 i.e. don’t know which states are good or what the actions do
 Must actually try actions and states out to learn

31. Passive Learning
 Simplified task
 You don’t know the transitions T(s,a,s’)
 You don’t know the rewards R(s,a,s’)
 You are given a policy (s)
 Goal: learn the state values
 … what policy evaluation did
 In this case:
 Learner “along for the ride”
 No choice about what actions to take
 Just execute the policy and learn from experience
 We’ll get to the active case soon
 This is NOT offline planning! You actually take actions in the
world and see what happens…
37

32. Example: Direct Evaluation
 Episodes:
(1,1) up -1
(1,2) up -1
(1,2) up -1
(1,3) right -1
(2,3) right -1
(3,3) right -1
(3,2) up -1
(3,3) right -1
(4,3) exit +100
(done)
x
y
(1,1) up -1
(1,2) up -1
(1,3) right -1
(2,3) right -1
(3,3) right -1
(3,2) up -1
(4,2) exit -100
(done)
V(2,3) ~ (96 + -103) / 2 = -3.5
V(3,3) ~ (99 + 97 + -102) / 3 = 31.3
= 1, R = -1
+100
-100
38

33. Recap: Model-Based Policy Evaluation
 Simplified Bellman updates to
calculate V for a fixed policy:
 New V is expected one-step-look-
ahead using current V
 Unfortunately, need T and R
s
(s)
s, (s)
s, (s),s’
s’
39

34. Model-Based Learning
 Idea:
 Learn the model empirically through experience
 Solve for values as if the learned model were correct
 Simple empirical model learning
 Count outcomes for each s,a
 Normalize to give estimate of T(s,a,s’)
 Discover R(s,a,s’) when we experience (s,a,s’)
 Solving the MDP with the learned model
 Iterative policy evaluation, for example
s
(s)
s, (s)
s, (s),s’
s’
40

35. Example: Model-Based Learning
 Episodes:
(1,1) up -1
(1,2) up -1
(1,2) up -1
(1,3) right -1
(2,3) right -1
(3,3) right -1
(3,2) up -1
(3,3) right -1
(4,3) exit +100
(done)
x
y
T(<3,3>, right, <4,3>) = 1 / 3
T(<2,3>, right, <3,3>) = 2 / 2
+100
-100
41
= 1
(1,1) up -1
(1,2) up -1
(1,3) right -1
(2,3) right -1
(3,3) right -1
(3,2) up -1
(4,2) exit -100
(done)

36. Model-Free Learning
 Want to compute an expectation weighted by P(x):
 Model-based: estimate P(x) from samples, compute expectation
 Model-free: estimate expectation directly from samples
 Why does this work? Because samples appear with the right
frequencies!
42

37. Sample-Based Policy Evaluation?
 Who needs T and R? Approximate the
expectation with samples (drawn from T!)
s
(s)
s, (s)
s, (s),s’
s2’ s3’
s1
’’
43
Almost! But we only
actually make progress
when we move to i+1.

38. Temporal-Difference Learning
 Big idea: learn from every experience!
 Update V(s) each time we experience (s,a,s’,r)
 Likely s’ will contribute updates more often
 Temporal difference learning
 Policy still fixed!
 Move values toward value of whatever
successor occurs: running average!
s
(s)
s, (s)
s’
Sample of V(s):
Update to V(s):
Same update:
44

39. Exponential Moving Average
 Exponential moving average
 Makes recent samples more important
 Forgets about the past (distant past values were wrong anyway)
 Easy to compute from the running average
 Decreasing learning rate can give converging averages
45

40. Example: TD Policy Evaluation
T
ake = 1,  = 0.5
(1,1) up -1
(1,2) up -1
(1,2) up -1
(1,3) right -1
(2,3) right -1
(3,3) right -1
(3,2) up -1
(3,3) right -1
(4,3) exit +100
(done)
46
(1,1) up -1
(1,2) up -1
(1,3) right -1
(2,3) right -1
(3,3) right -1
(3,2) up -1
(4,2) exit -100
(done)

41. Problems with TD Value Learning
 TD value leaning is a model-free way
to do policy evaluation
 However, if we want to turn values into
a (new) policy, we’re sunk:
 Idea: learn Q-values directly
 Makes action selection model-free too!
s
a
s, a
s,a,s’
s’
47

42. Active Learning
 Full reinforcement learning
 You don’t know the transitions T(s,a,s’)
 You don’t know the rewards R(s,a,s’)
 You can choose any actions you like
 Goal: learn the optimal policy
 … what value iteration did!
 In this case:
 Learner makes choices!
 Fundamental tradeoff: exploration vs. exploitation
 This is NOT offline planning! You actually take actions in the
world and find out what happens…
48

43. The Story So Far: MDPs and RL
 Compute V*, Q*, * exactly 
 Evaluate a fixed policy 
 If we don’t know the MDP
 We can estimate the MDP then solve
 We can estimate V for a fixed policy 
 We can estimate Q*(s,a) for the
optimal policy while executing an
exploration policy
Value and policy
Iteration
 Policy evaluation
Things we know how to do: Techniques:
 If we know the MDP  Model-based DPs
 Model-based RL
 Model-free RL:
 Value learning
 Q-learning
49

44. Q-Learning
 Q-Learning: sample-based Q-value iteration
 Learn Q*(s,a) values
 Receive a sample (s,a,s’,r)
 Consider your old estimate:
 Consider your new sample estimate:
 Incorporate the new estimate into a running average:
52

45. Q-Learning Properties
 Amazing result: Q-learning converges to optimal policy
 If you explore enough
 If you make the learning rate small enough
 … but not decrease it too quickly!
 Basically doesn’t matter how you select actions (!)
 Neat property: off-policy learning
 learn optimal policy without following it (some caveats)
S E S E
53

46. Exploration / Exploitation
 Several schemes for forcing exploration
 Simplest: random actions ( greedy)
 Every time step, flip a coin
 With probability , act randomly
 With probability 1-, act according to current policy
 Problems with random actions?
 You do explore the space, but keep thrashing
around once learning is done
 One solution: lower  over time
 Another solution: exploration functions
54

47. Exploration Functions
 When to explore
 Random actions: explore a fixed amount
 Better idea: explore areas whose badness is not (yet)
established
 Exploration function
 Takes a value estimate and a count, and returns an optimistic
utility, e.g. (exact form not important)
55

48. Q-Learning
 Q-learning produces tables of q-values:
56

49. Q-Learning
57
 In realistic situations, we cannot possibly learn
about every single state!
 Too many states to visit them all in training
 Too many states to hold the q-tables in memory
 Instead, we want to generalize:
 Learn about some small number of training states
from experience
 Generalize that experience to new, similar states
 This is a fundamental idea in machine learning, and
we’ll see it over and over again

50. Example: Pacman
 Let’s say we discover
through experience
that this state is bad:
 In naïve q learning, we
know nothing about
this state or its q
states:
 Or even this one!
58

51. Feature-Based Representations
 Solution: describe a state using
a vector of features
 Features are functions from states
to real numbers (often 0/1) that
capture important properties of the
state
 Example features:
 Distance to closest ghost
 Distance to closest dot
 Number of ghosts
 1 / (dist to dot)2
 Is Pacman in a tunnel? (0/1)
 …… etc.
 Can also describe a q-state (s, a)
with features (e.g. action moves
closer to food)
59

52. Linear Feature Functions
 Using a feature representation, we can write a
q function (or value function) for any state
using a few weights:
 Advantage: our experience is summed up in a
few powerful numbers
 Disadvantage: states may share features but
be very different in value!
60

53. Function Approximation
 Q-learning with linear q-functions:
 Intuitive interpretation:
 Adjust weights of active features
 E.g. if something unexpectedly bad happens, disprefer all states
with that state’s features
 Formal justification: online least squares
61

54. Example: Q-Pacman
62

55. Policy Search
69
http://heli.stanford.edu/

56. Policy Search
70
 Problem: often the feature-based policies that work well
aren’t the ones that approximate V / Q best
 E.g. your value functions from project 2 were probably horrible
estimates of future rewards, but they still produced good
decisions
 We’ll see this distinction between modeling and prediction again
later in the course
 Solution: learn the policy that maximizes rewards rather
than the value that predicts rewards
 This is the idea behind policy search, such as what
controlled the upside-down helicopter

57. Policy Search
71
 Simplest policy search:
 Start with an initial linear value function or q-function
 Nudge each feature weight up and down and see if
your policy is better than before
 Problems:
 How do we tell the policy got better?
 Need to run many sample episodes!
 If there are a lot of features, this can be impractical

58. Policy Search*
 Advanced policy search:
 Write a stochastic (soft) policy:
 Turns out you can efficiently approximate the
derivative of the returns with respect to the
parameters w (details in the book, but you don’t have
to know them)
 Take uphill steps, recalculate derivatives, etc.
72